Using rationality, like in language theory, we define a family of infinite graphs. This family is a strict extension of the context-free graphs of Muller and Schupp, the equational graphs of Courcelle and the prefix recognizable graphs of Caucal. We give basic properties, as well as an internal and an external characterization of these graphs. We also show that their traces form an AFL of recursive languages, containing the context-free languages.

Diagnosis problems of discrete-event systems consist in detecting unobservable defects during system execution. For finite-state systems, the theory is well understood and a number of effective solutions have been developed. For infinite-state systems, however, there are only few results, mostly identifying classes where the problem is undecidable.We consider higher-order pushdown systems and investigate two basic variants of diagnosis problems: the diagnosability, which consists in deciding whether defects can be detected within a finite delay, and the bounded-latency problem, which consists in determining a bound for the delay of detecting defects.We establish that the diagnosability problem is decidable for arbitrary sub-classes of higher-order visibly pushdown systems provided unobservable events leave the stacks unchanged. For this case, we present an effective algorithm. Otherwise, we show that diagnosability becomes undecidable already for first-order visibly pushdown automata. Furthermore, we establish that the bounded-latency problem for higher-order pushdown systems is as hard as deciding finiteness of a higher-order pushdown language. This is in contrast with the case of finite-state systems where the problem reduces to diagnosability.Key-words: Pushdown systems, Visibly pushdown systems, Partial observation, Diagnosis. Diagnostic des Systèmesà PilesRésumé : Les problèmes de diagnostic des systèmesàévénements discrets examinent la détection de défauts inobservables au cours de l'exécution du système. Pour les systèmesà nombre d'états fini, la théorie est déjà bien maîtrisée, et de nombreuses solutions effectives ontété développées. En revanche, le cas des systèmesà nombre d'états infini n'a fait l'objet que de peu d'études, exhibant le plus souvent des problèmes indécidables. Nous considérons les systèmesà piles d'ordre supérieur etétudions deux problèmeś elémentaires de diagnostic : la diagnosticabilité, problème pour lequel il s'agit de décider si les défauts seront détectés en un temps fini, et le problème de la latence bornée, pour lequel on souhaite déterminer si le décalage temporel entre l'occurrence d'un défaut et sa détection est borné.Nousétablissons que la diagnosticabilité est décidable pour toute sous-classe de systèmes a piles d'ordre supérieur dès lors que que toutévénement inobservable ne modifie l'état des piles. Dans ce cas, nous présentons des algorithmes. Autrement, nous montrons que la diagnosticabilité est indécidable pour la sous-classe des systèmes dits "visibly pushdown", un classique de la littérature. En outre, nousétablissons que le problème de la latence bornée pour les systèmesà piles d'ordre supérieur est au moins aussi difficile que celui de décider la finitude des langagesà piles d'ordre supérieur. Ce dernier résultat est en opposition avec le cas des systèmesà nombre d'états fini pour lesquels le problème de la latence bornée se réduità celui de diagnosticabilité.

Abstract. This paper shows that the traces of rational graphs coincide with the context-sensitive languages.

The analysis of discrete event systems under partial observation is an important topic, with major applications such as the detection of information flow and the diagnosis of faulty behaviors. These questions have, mostly, not been addressed for classical models of recursive systems, such as pushdown systems and recursive state machines. In this paper, we consider recursive tile systems, which are recursive infinite systems generated by a finite collection of finite tiles, a simplified variant of deterministic graph grammars (slightly more general than pushdown systems). Since these systems are infinite-state in general powerset constructions for monitoring do not always apply. We exhibit computable conditions on recursive tile systems and present non-trivial constructions that yield effective computation of the monitors. We apply these results to the classic problems of state-based opacity and diagnosability (off-line verification of opacity and diagnosability, and also runtime monitoring of these properties). For a decidable subclass of recursive tile systems, we also establish the decidability of the problems of state-based opacity and diagnosability.